Optimal. Leaf size=65 \[ \frac{a c^2 \sqrt{c x^2} (d x)^{m+6}}{d^6 (m+6) x}+\frac{b c^2 \sqrt{c x^2} (d x)^{m+7}}{d^7 (m+7) x} \]
[Out]
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Rubi [A] time = 0.0711002, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15 \[ \frac{a c^2 \sqrt{c x^2} (d x)^{m+6}}{d^6 (m+6) x}+\frac{b c^2 \sqrt{c x^2} (d x)^{m+7}}{d^7 (m+7) x} \]
Antiderivative was successfully verified.
[In] Int[(d*x)^m*(c*x^2)^(5/2)*(a + b*x),x]
[Out]
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Rubi in Sympy [A] time = 17.9647, size = 56, normalized size = 0.86 \[ \frac{a c^{2} \sqrt{c x^{2}} \left (d x\right )^{m + 6}}{d^{6} x \left (m + 6\right )} + \frac{b c^{2} \sqrt{c x^{2}} \left (d x\right )^{m + 7}}{d^{7} x \left (m + 7\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((d*x)**m*(c*x**2)**(5/2)*(b*x+a),x)
[Out]
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Mathematica [A] time = 0.0361043, size = 39, normalized size = 0.6 \[ \frac{\left (c x^2\right )^{5/2} (d x)^m \left (\frac{a x^6}{m+6}+\frac{b x^7}{m+7}\right )}{x^5} \]
Antiderivative was successfully verified.
[In] Integrate[(d*x)^m*(c*x^2)^(5/2)*(a + b*x),x]
[Out]
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Maple [A] time = 0.004, size = 40, normalized size = 0.6 \[{\frac{ \left ( bmx+am+6\,bx+7\,a \right ) x \left ( dx \right ) ^{m}}{ \left ( 7+m \right ) \left ( 6+m \right ) } \left ( c{x}^{2} \right ) ^{{\frac{5}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((d*x)^m*(c*x^2)^(5/2)*(b*x+a),x)
[Out]
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Maxima [A] time = 1.3691, size = 53, normalized size = 0.82 \[ \frac{b c^{\frac{5}{2}} d^{m} x^{7} x^{m}}{m + 7} + \frac{a c^{\frac{5}{2}} d^{m} x^{6} x^{m}}{m + 6} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2)^(5/2)*(b*x + a)*(d*x)^m,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.231621, size = 78, normalized size = 1.2 \[ \frac{{\left ({\left (b c^{2} m + 6 \, b c^{2}\right )} x^{6} +{\left (a c^{2} m + 7 \, a c^{2}\right )} x^{5}\right )} \sqrt{c x^{2}} \left (d x\right )^{m}}{m^{2} + 13 \, m + 42} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2)^(5/2)*(b*x + a)*(d*x)^m,x, algorithm="fricas")
[Out]
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x)**m*(c*x**2)**(5/2)*(b*x+a),x)
[Out]
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GIAC/XCAS [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2)^(5/2)*(b*x + a)*(d*x)^m,x, algorithm="giac")
[Out]